1. Two firms bid for a contract to build a university building. Their construction costs are independent and uniformly drawn from [0, 1]. Both bidders submit their bids si- multaneously. The winner is the bidder who submits a lowest bid. Tie-breaking rule is random. This kind of bidding game is called "reverse auction" because the bidders bid for the right to provide a service and the winner is paid for the service. The FCC in 2017 has adopted similar auctions designed to repurpose spectrum for new uses. (a) In the first auction, the winner gets paid the loser's bid. For example, if the winner's cost is 0.5, his bid is 0.56, and the loser's bid is 0.6, then the winner gets the contract and the university pays the winner 0.6. The winner's net profit from the contract is 0.6-0.5 0.1. Solve for a Bayesian Nash equilibrium of this auction. What is the equilibrium bid of a firm bid if his cost is actually 0.5? (b) In the second auction, the winner gets paid the his/her own winning bid. For example, if the winner's cost is 0.5, his bid is 0.56, and the loser's bid is 0.6, then the winner gets the contract and is paid a price of 0.56. The winner's net profit is 0.56 0.5 0.06. Analyze this auction. =